Intermediate Value Theorem Examples

Example 1

Can we use the IVT to conclude that f(x) = sin(x) passes through y = 0 on ?

Yes. The function f is continuous on the closed interval . We have and , therefore and f(b) = 1. Since f(a) = -1 < 0 < 1 = f(b),
the IVT says there is some c in between a and b (that is, ), f(c) = 0.

Example 2

Can we use the IVT to conclude that passes through y = 1 on ?

No. We cannot use the IVT in this case because the function f is discontinuous at x = 0 and not continuous on

Example 3

Can we use the IVT to conclude that passes through y = 1 on (0,1)?

No. This question doesn't even make sense. In order to use the IVT we need to know the function values at the endpoints of the interval, but f(0) is undefined.

Example 4

Can we use the IVT to conclude that f(x) = x2 passes through y = 0 on (-1,1)?

No. Since f(a) = (-1) 2 = 1 and f(b) = (1) 2 = 1, 0 is not between f(a) and f(b). We cannot use the IVT in this situation. The function f does in fact pass throughy = 0, but we know that by observation, not by using the IVT.

Reminder: It doesn't matter whether we have f(a) < M < f(b) or f(b) < M < f(a), as long as M is between f(a) and f(b).